Suppose that we roll a red and a black die. let a = "the black die shows a 2 or a 5", b = "the sum of the two dice is at least 7". are events a and b independent?

No, A and B are not independent events.

∵ it does not satisfy the rule of probability for independent events i.e.

P(A∩B)=P(A).P(B)

Explanation:

Let A be the event that  the black dice shows 2 or 5

Let B be the event that the sum of two dice is atleast 7

Sample space of A={(}

Sample space of B= { ,, }

P(A)=

⇒P(A)=

⇒P(A)=

Similarly,

P(B)=

⇒ P(B) =

Now for Sample Space of (A∩B)= {}

So, P(A∩B)=

Now we apply the formula,

P(A).P(B)=P(A∩B)

× ≠

≠

∴ The events A and B are not independent events.

step-by-step explanation: